Question

If \(z=14-4 x y\) evaluate \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) at the point \((1,2)\).

Short answer

Question: Find the partial derivatives of the function \(z = 14-4xy\) with respect to x and y, and evaluate them at the point (1, 2). Answer: The partial derivatives of the function z at the point (1, 2) are: \(\frac{\partial z}{\partial x}\bigg|_{(1,2)} = -8\) and \(\frac{\partial z}{\partial y}\bigg|_{(1,2)} = -4\).

Step-by-step solution

Find the partial derivative with respect to x

To find the partial derivative of z with respect to x, we will treat y as a constant and differentiate \(14 - 4xy\) with respect to x: \[\frac{\partial z}{\partial x} = -4y\]

Find the partial derivative with respect to y

Similarly, to find the partial derivative of z with respect to y, we will treat x as a constant and differentiate \(14 - 4xy\) with respect to y: \[\frac{\partial z}{\partial y} = -4x\]

Evaluate the partial derivatives at the point (1, 2)

Now, we plug in the values of the point (1, 2) into our partial derivatives: \[\frac{\partial z}{\partial x}\bigg|_{(1,2)} = -4(2) = -8\] \[\frac{\partial z}{\partial y}\bigg|_{(1,2)} = -4(1) = -4\] So, the partial derivatives of z with respect to x and y at the point (1, 2) are -8 and -4, respectively.

Key concepts

Multivariable Calculus

In the realm of mathematics, multivariable calculus is an extension of calculus that deals with functions involving more than one variable. It's like entering a room where, instead of focusing on one object, we now have to pay attention to several objects at once and understand how they all interact with each other. Imagine you're in a situation where the temperature in a room changes depending on where you're standing and the time of day. Multivariable calculus allows us to study these changes and understand the relationships between different variables involved.

Understanding multivariable functions and their changes is crucial because it opens up the world of 3D space and beyond, where many real-world phenomena like weather patterns, fluid dynamics, and economic models live. The function from our exercise, for example, involves two variables, x and y, which means that changes in z can depend on changes in both of these variables at the same time. Through this lens, we can begin to explore the complex interplay between different factors in multivariable systems.

Partial Differentiation

When we deal with functions of multiple variables, partial differentiation comes into play as a specialized tool. Here's a simple way to think about it: if you're responsible for watering plants in a large nursery, you can't be everywhere at once, so you focus on watering one row at a time. Similarly, when we perform partial differentiation, we concentrate on how a function changes with respect to one variable at a time, keeping all others constant.

When you see notation like \( \frac{\partial z}{\partial x} \) or \( \frac{\partial z}{\partial y} \) in our exercise, it's like saying, 'Let's see how z changes if I only move along the x-axis or the y-axis and ignore the other.' This concept is essential for understanding how complex systems change in one direction, isolating the effect of one variable while holding others constant. It's a fundamental tool in optimizing and controlling systems in fields like engineering, physics, and economics.

Evaluating Derivatives at a Point

Now that we've computed the partial derivatives, a natural next step is to make those derivatives concrete by evaluating them at a specific point. This is called evaluating derivatives at a point. Think of it as zeroing in on a specific location in our nursery and checking exactly how much water pressure you'll need for the plants right there.

In the context of our exercise, after finding the general expressions for the rate of change of z with respect to x and y, we evaluated these expressions at the point (1, 2). What we ended up with - -8 for \( \frac{\partial z}{\partial x} \) and -4 for \( \frac{\partial z}{\partial y} \) - are like instant snapshots of how z responds to tiny nudges in the x and y directions at that exact spot. These values can help predict the behavior of z near the point (1, 2) and are crucial in applications such as gradient descent algorithms in machine learning, where one needs to know the direction and rate of steepest descent at a given point.